CNRS researcher at the university of Paris 13.

I am interested in algebraic topology, algebraic geometry, and anything in between.

CNRS researcher at the university of Paris 13.

I am interested in algebraic topology, algebraic geometry, and anything in between.

The goal of this project is to develop real algebraic K-theory for * Poincaré ∞-categories*, that is, stable ∞-categories equipped with a non-degenerate quadratic functor. This yields a genuine C₂-equivariant theory whose underlying theory is the algebraic K-theory of the underlying stable ∞-category and whose geometric fixed points gives L-theory (as developped in the ∞-categorical setting by Lurie). When the ∞-category in question is that of perfect complexes over a ring, a suitable choice of a quadratic functor reproduces the L-theory studied by Ranicki in his seminal work on surgery theory. In fact, our approach is strongly inspired by surgery theory, and we work closely with the *bordism ∞-category* associated to a Poincaré ∞-category. The genuine fixed points of our theory yield a Grothendieck-Witt type theory for Poincaré ∞-categories, which enjoys a variety of favorable formal properties, including additivity for quasi-split verdier sequences and a universal characterization in these terms. We also give a universal characterization of the resulting L-theory in terms of a suitable notion of bordism invariance. We expect this theory to yields a very convenient framework to "do Hermitian K-theory" when 2 is not invertible.

Higher Massey products are generalizations of ordinary cup products which can be used to detect the higher coherences of homotopy coherent multiplicative structures. When applied to the multiplicative structure underlying Galois cohomology these higher Massey products are conjectured to vanish. Interestingly enough, this conjecture is equivalent to the existence of rational points on certain homogeneous spaces whose geometric stabilizers are finite groups of unipotent type. In this project we use recent joint work concerning rational points on such homogeneous spaces in order to prove the Massey vanishing conjecture for all number fields.

This project concerns the Quillen cohomology of higher categories together with the closely related notion of small extensions. In previous work we have shown that the Quillen cohomology of an ∞-category can be phrased in terms of its *twisted arrow category*, while the corresponding result for (∞,2)-categories leads to a new notion of the *twisted 2-cell category*. We are currently investigating further the notion of small extension of ∞-categories and the role it plays in Postnikov type decompositions, in deformation theory, and in Goerss-Hopkins style obstruction theories.

This project is dedicated to establishing various foundational aspects of the theory of (∞,2)-categories, using the model of scaled simplicial sets. Topics include Cartesian and opCartesian fibrations, the Gray product as a left Quillen functors and a basic theory of limits and colimits. We also prove a technical result stating that the notion of a weak ∞-bicategory (which is defined in terms analogous to the horn filling conditions in quasi-categories) is equivalent to that of an ∞-bicategory, i.e., a fibrant scaled simplicial set.

#### The Massey vanishing conjecture for number fields

## with Olivier Wittenberg,

*submitted*(pdf, arXiv).#### Lax limits of model categories

*submitted*(pdf, arXiv).- Appendix to
#### Number fields with prescribed norms (C. Frei, D. Loughran and R. Newton)

## with Olivier Wittenberg,

*submitted*(arXiv). #### Zéro-cycles sur les espaces homogènes et problème de Galois inverse

## with Olivier Wittenberg,

*submitted*(pdf, arXiv).#### Ambidexterity and the universality of finite spans

*submitted*(pdf, arXiv).#### The tangent bundle of a model category

## with Joost Nuiten and Matan Prasma,

*Submitted*(pdf, arXiv).#### Quillen cohomology of (∞,2)-categories

## with Joost Nuiten and Matan Prasma,

*Higher Structures*, 3 (1), 2019, p. 17–66, (pdf, arXiv).#### Tangent categories of algebras over operads

## with Joost Nuiten and Matan Prasma,

*The Israel Journal of Mathematics*, to appear (pdf, arXiv).#### Second descent and rational points on Kummer varieties

*Proceedings of the London Mathematical Society*, to appear (pdf, arXiv).#### The abstract cotangent complex and Quillen cohomology of enriched categories

## with Joost Nuiten and Matan Prasma,

*Journal of Topology*, 11 (3), 2018, p. 752-798 (pdf, arXiv).#### Geometry and arithmetic of certain log K3 surface

*Annales de l'Institut Fourier*, to appear (pdf, arXiv).#### Integral points on conic log K3 surfaces

*Journal of the European Mathematical Society*, to appear (pdf, arXiv).#### Pro-categories in homotopy theory

## with Ilan Barnea and Geoffroy Horel,

*Algebraic and Geometric Topology*, 17 (1), 2017, p. 567-643 (pdf, arXiv).#### An integral model structure and truncation theory for coherent group actions

## with Matan Prasma,

*The Israel Journal of Mathematics*, to appear (pdf, arXiv).#### Hasse principle for Kummer varieties

## with Alexei Skorobogatov,

*Algebra and Number Theory*, 10.4, 2016, p. 813-841 (pdf, arXiv).#### On the fibration method for zero-cycles and rational points

## with Olivier Wittenberg,

*Annals of Mathematics*, 183 (1), 2015, p. 229-295 (pdf, arXiv).#### The Grothendieck construction for model categories

## with Matan Prasma,

*Advances in Mathematics*, 281, 2015, p. 1306-1363 (pdf, arXiv).#### Quasi-unital ∞-categories.

*Algebraic and Geometric Topology*, 15 (4), 2015, p. 2303-2381 (pdf, arXiv).#### The Hardy-Littlewood conjecture and rational points

## with Alexei Skorobogatov and Olivier Wittenberg,

*Compositio Mathematica*, 150, 2014, p. 2095-2111 (pdf, arXiv).#### Singular curves and the étale Brauer-Manin obstruction for surfaces

## with Alexei Skorobogatov,

*Annales Scientifiques de l'École Normale Supérieure*, 47, 2014, p. 765-778 (pdf, arXiv).#### Homotopy obstructions to rational points

## with Tomer Schlank, In: Alexei Skorobogatov (Ed.), Torsors, Étale Homotopy and Applications to Rational Points, LMS Lecture Notes Series 405, Cambridge University Press, 2013, pp. 280-413 (pdf, arXiv).

#### The cobordism hypothesis in dimension 1

## We give a proof of the cobordism hypothesis in dimension 1 using the theory of quasi-unital ∞-categories (pdf, arXiv).

#### The section conjecture for graphs and conical curves

## We show that the finite descent obstruction controls the existence of rational points on normal crossing singular curves whose components are all of genus 0, by relating the problem to a fix point property of pro-finite groups acting on pro-finite trees (pdf, arXiv).

#### Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Bordeaux, March 2018.

#### Zéro-cycles sur les espaces homogènes et problème de Galois inverse, séminaire théorie de nombres de Jussieu, March 2018.

#### Ambidexterity and the universality of finite spans, séminaire de topologie, université de Lille, January 2018.

#### Cohomology of higher categories, réunion annuelle du GdR topologie algébrique et applications, October 2017 (pdf).

#### Quillen obstruction theory, séminaire homotopie en géométrie algébrique, October 2017 (pdf).

#### Zero-cycles on homogeneous spaces, Rational points 2017, Franken-Akademie Schloss Schney, July 2017.

#### Quillen cohomology of enriched categories, séminaire de topologie, géométrie et algèbre, Nantes, June 2017 (pdf).

#### Deuxième descente et points rationnels sur les surfaces de Kummer, séminaire arithmétique et géométrie algébrique, université de Strasbourg, January 2017.

#### Second 2-descent and rational points on Kummer surfaces, Rational points and algebraic geometry, Luminy September 2016 (pdf).

#### Higher additivity, higher monoids and the universal property of finite spans., Homotopy Theory Day, Hebrew University of Jerusalem, July 2016 (pdf).

#### Abstract representation theory and the cotangent complex formalism, congrès national de la Société Mathématique de France, Université François Rabelais, Tours, June 2016 (pdf).

#### The cotangent complex formalism, séminaire de physique mathématique et de topologie algébrique, université de Angers, May 2016 (pdf).

#### Pro-categories in homotopy theory, séminaire de Topologie, Université de Lille, April 2016 (pdf).

#### Integral Points on log K3 surfaces, séminaire Variétés Rationnelles, ENS, Paris, April 2016 (pdf).

#### The cobordism hypothesis in dimension 1, séminaire de topologie, géométrie et algèbre, université de Nantes, March 2016.

#### Integral Points on log K3 surfaces, séminaire d'arithmétique et de géométrie algébrique, université Paris-Sud, March 2016 (pdf).

#### Pro-categories in homotopy theory, séminaire de topologie algébrique, université Paris 13, January 2016 (pdf).

#### Rational points on fibered varieties, Interactions between arithmetic and homotopy, Royal Imperial College, London, September 2015 (pdf).

#### The Hasse principle for generalized Kummer varieties, Göttingen-Hannover number theory workshop, Leibniz university of Hannover, July 2015.

#### The descent-fibration method for integral points, Arithmetic geometry, Chow groups and rational points, The Euler Institute, St. Petersburg, June 2015 (pdf).

#### The descent-fibration method for integral points, Heilbronn seminar, university of Bristol, Bristol, May 2015 (pdf).

#### On the fibration method, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, February 2015 (pdf).

#### Model fibrations and the Grothendieck correspondence for model categories, Higher Geometric Structures along the Lower Rhine V, Radboud university Nijmegen, June 2014.

#### From linear equations in primes to the fibration method, The intercity number theory seminar, university of Groningen, October 2013.

#### The section conjecture for graphs and applications for singular curves, London-Paris number theory seminar, Royal Imperial College, London, June 2013.

#### The Hasse principle for singular curves with applications for smooth surfaces, séminaire Variétés Rationnelles, Institut Henri Poincaré, Paris, March 2013.

#### Quasi-unital ∞-categories and the cobordism hypothesis in dimension 1, algebraic topology seminar, Radboud university Nijmegen, November 2012 (pdf)

#### Étale homotopy theory, Workshop on Cohomological Methods in Arithmetic Geometry, institut für Mathematik, Zurich, September 2012 (pdf)

#### Étale homotopy and Diophantine equations, Workshop on Arithmetic Geometry and Homotopy Theory, Imperial College, London, May 2012 (pdf)

#### Homotopy obstructions to rational points, joint mathematics meeting, AMS Special Session on Rational Points on Varieties, Boston, January 2012 (pdf)

#### Homotopy obstructions to integral points, Torsors: theory and applications, International Centre for Mathematical Sciences, Edinburgh, January 2011 (pdf)

#### The Étale Homotopy Type and the Local-Global Principle, Rational Points 3, Universität Bayreuth, July 2010 (pdf).

#### Rational points on elliptic surfaces, notes for minicourse given at the IHP, spring 2019 (pdf).

#### Little cube algebras and factorization homology, notes for master course given in Paris 13, spring 2019 (draft under construction, last update 16/04/2019, pdf).

#### Topological Hochschild homology as a cyclotomic spectrum, Groupe de travail sur THH, Université de Paris 6, 2018 (pdf).

#### Factorization homology for topological manifolds, Groupe de travail sur les homologie de factorization, Université de Paris 13, 2017 (pdf).

#### Introduction to stable ∞-categories, Groupe de travail sur les ∞-catégories, Université de Paris 13, 2017 (pdf).

#### (co)Cartesian fibrations, Caesarea 2016 (pdf).

#### Arizona Winter School (2015)

#### Simplicial homotopy theory, Master course, Radboud university 2014 (pdf).

#### Limits, colimits and adjunctions in ∞-categories, Workshop on ∞-Categories, Université Catholique de Louvain, 2013 (pdf).

#### 2-fold complete Segal spaces, Caesarea 2013 (pdf).

#### Étale homotopy and Diophantine equations, Minicourse, Bernoulli centre, EPFL 2012 (link).

#### General relativity, HUJI 2011 (pdf).

#### Basic notions in algebraic topology, TA notes, HUJI 2009-2011 (pdf).

#### ∞-sheaves, Caesarea 2011 (pdf).

#### Elliptic regularity, HUJI 2010 (pdf).

#### Algebraic structures 1, TA notes, HUJI fall 2010 (pdf).

#### Quasi-categories, Caesarea 2010 (pdf).

#### Basic notions in differentials geometry, TA notes, HUJI spring 2009 (pdf).

#### Complex cobordism and formal group laws, Caesarea 2009 (pdf).

#### Classification of framed manifolds, MIT Kan seminar 2009 (pdf).

#### Loop structures on the 3-sphere, MIT babytop seminar 2009 (pdf).

#### Curves, surfaces and the Hasse principle, Summer School on Rational Points, HUJI 2009 (day 1,day 2, day 3).

#### Periodicity in stable homotopy theory 1, HUJI student seminar 2009 (pdf).

#### Periodicity in stable homotopy theory 2, HUJI student seminar 2009 (pdf).

#### Polyhedral groups, HUJI 2009 (pdf).

#### The classification problem for smooth manifolds, HUJI student seminar 2008 (pdf).

#### Galois cohomology and elliptic curves, HUJI 2008 (pdf).

#### Modular curves and modular forms, HUJI 2008 (pdf).